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❆❧❣❡❜r❛✐❝ ❣❡♦♠❡tr② ♦✈❡r ❢r❡❡ ❧❡❢t r❡❣✉❧❛r ❜❛♥❞ ❆rt❡♠ ❙❤❡✈❧②❛❦♦✈ ✸✵t❤ ❆♣r✐❧

■♥❢♦r♠❛❧ ❞❡✜♥✐t✐♦♥ ▲❡t A ∗ ❜❡ t❤❡ ❢r❡❡ s❡♠✐❣r♦✉♣ ♦❢ ❛♥ ❛❧♣❤❛❜❡t A = { ❛❧❧ ❘✉ss✐❛♥ ❧❡tt❡rs } ✳ 1 ⊆ A ∗ ❜❡ t❤❡ s❡t ♦❢ ✇♦r❞s ❆♥② ❘✉ss✐❛♥ ✇♦r❞ ✐s ❛♥ ❡❧❡♠❡♥t ♦❢ A ∗ ✳ ▲❡t A ∗ ✇✐t❤ ♥♦ r❡♣❡t✐t✐♦♥s✳ ■s t❤❡ s❡t A ∗ 1 ✐♥t❡r❡st✐♥❣❄

▼❛♥② ♣♦♣✉❧❛r ✭s❡❡ ❚❱✮ ✇♦r❞s ❞♦ ♥♦t ❝♦♥t❛✐♥ ❧❡tt❡r r❡♣❡t✐t✐♦♥s✿ Ïóòèí✱ âîäêà✱ êðûìíàø✱ Ñòàëèí✱ íåôòü✱ áàíäåðîâöû✱ ✃àäûðîâ✱ ðóáëü✱ äåìîêðàòèÿ✱ ➹åéðîïà✱ äóõîâíûý ñêðåïû✱ âàòíèê✳

❆♠❛③✐♥❣❧②✱ t❤❡r❡ ❡①✐st s❡♠✐❣r♦✉♣s t❤❛t ❞❡s❝r✐❜❡ ✇♦r❞s ✇✐t❤ ♥♦ ❧❡tt❡r r❡♣❡t✐t✐♦♥s✳ ❆ s❡♠✐❣r♦✉♣ ✐s ❝❛❧❧❡❞ ❛ ❧❡❢t r❡❣✉❧❛r ❜❛♥❞ ✭▲❘❇✮ ✐❢ t❤❡ ✐❞❡♥t✐t✐❡s xx = x ✱ xyx = xy ❤♦❧❞✳

❉❡✜♥✐t✐♦♥ ❚❤❡ ❢r❡❡ ▲❘❇ F n ♦❢ r❛♥❦ n ✐s t❤❡ s❡t ♦❢ ❛❧❧ ✇♦r❞s ♦❢ t❤❡ ❛❧♣❤❛❜❡t { a 1 , a 2 , . . . , a n } s✉❝❤ t❤❛t ❡❛❝❤ ✇♦r❞ ❝♦♥s✐sts ♦❢ ❞✐✛❡r❡♥t ❧❡tt❡rs✳ ❊✳❣✳ F 3 = { a 1 , a 2 , a 3 , a 1 a 2 , a 2 a 1 , a 1 a 3 , a 3 a 1 , a 3 a 2 , a 2 a 3 , a 1 a 2 a 3 , a 1 a 3 a 2 , a 2 a 1 a 3 , a 2 a 3 a 1 , a 3 a 1 a 2 , a 3 a 2 a 1 } ❚❤❡ ♣r♦❞✉❝t ♦❢ w 1 , w 2 ∈ F n ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✿ w 1 ◦ w 2 = w 1 ( w 2 ) ∃ , ✇❤❡r❡ t❤❡ ♦♣❡r❛t♦r ∃ ❞❡❧❡t❡s ❛❧❧ ❧❡tt❡rs ♦❢ w 2 ✇❤✐❝❤ ♦❝❝✉r ❡❛r❧✐❡r✳ ❋♦r ❡①❛♠♣❧❡✱ ( a 1 a 2 )( a 2 a 3 a 1 ) = a 1 a 2 a 3 . ❖❜✈✐♦✉s❧②✱ ❡❧❡♠❡♥ts ❝♦♥t❛✐♥✐♥❣ ❛❧❧ ❧❡tt❡rs ❛r❡ ❧❡❢t ③❡r♦❡s✿ ( a 1 a 2 a 3 ) x = a 1 a 2 a 3 ❢♦r ❛❧❧ x.

P❛rt✐❛❧ ♦r❞❡rs ♦✈❡r ▲❘❇✲s x ≤ y ⇔ xy = y ❋♦r ❡❧❡♠❡♥ts ♦❢ F n x ≤ y ♠❡❛♥s t❤❛t x ✐s ❛ ♣r❡✜① ♦❢ y ✳ ❊✳❣✳ a 1 a 2 ≤ a 1 a 2 a 3 , a 1 ≤ a 1 . ≤ ✲❝♦♠♣❛r❛❜❧❡ ❡❧❡♠❡♥ts ❛r❡ ❛❧✇❛②s ❝♦♠♠✉t❡✳

≤ ✲♦r❞❡r ♦✈❡r F n ✐s ❛ tr❡❡ ▲❡t ✉s ❛❞❥♦✐♥t ❛♥ ✐❞❡♥t✐t② ε t♦ F n ✱ ❤❡♥❝❡ t❤❡ ❍❛ss❡ ❞✐❛❣r❛♠ ♦❢ ≤ ✐s a 1 a 2 a 3 a 1 a 3 a 2 a 2 a 1 a 3 a 2 a 3 a 1 a 3 a 1 a 2 a 3 a 2 a 1 a 1 a 2 a 1 a 3 ❆ ✁ a 2 a 1 a 2 a 3 a 3 a 1 a 3 a 2 ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ a 1 ❅ ❆ ✁ ❆ ✁ ✏✏✏✏✏✏✏✏✏✏✏✏ a 2 a 3 ❅ ✁ ❅ ✁ ❅ ✁ ❅ ✁ ε

❘❛♥❞♦♠ ✇❛❧❦s ❚s❡t❧✐♥ ❧✐❜r❛r②✿ ❛ s❤❡❧❢ ♦❢ ❜♦♦❦s✳ ❆♥② ❜♦♦❦ ❤❛s ❛ ♣r♦❜❛❜✐❧✐t② p i ✳ ❲✐t❤ t❤❡ ❣✐✈❡♥ ♣r♦❜❛❜✐❧✐t✐❡s ✇❡ ❝❤♦♦s❡ ❛ ❜♦♦❦ ❛♥❞ ♣✉t ✐t ❛t t❤❡ ❢r♦♥t ♦❢ t❤❡ s❤❡❧❢✳ ■♥ ✐s ❛ r❛♥❞♦♠ ✇❛❧❦ ♦♥ t❤❡ ❈❛②❧❡② ❣r❛♣❤ ♦❢ ❢r❡❡ ▲❘❇ F n ✱ ✇❤❡r❡ n ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❜♦♦❦s✳ ❚❤❡ tr❛♥s✐t✐♦♥ ♦❢ t❤❡ ✇❛❧❦ ✐s ❧❡❢t ♠✉❧t✐♣❧✐❝❛t✐♦♥s✳ ❙❡❡ ❑✳ ❙✳ ❇r♦✇♥✱ ❙❡♠✐❣r♦✉♣s✱ r✐♥❣s✱ ❛♥❞ ▼❛r❦♦✈ ❝❤❛✐♥s✱ ❏✳ ❚❤❡♦r❡t✳ Pr♦❜❛❜✳✱ ✭✷✵✵✵✮✱ ✶✸✭✸✮✱ ✽✼✶✕✾✸✽✳ ❢♦r ❛♥♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ▲❘❇ ✐♥ r❛♥❞♦♠ ✇❛❧❦s✳

▼♦t✐✈❛t✐♦♥✿ ♠❛tr♦✐❞s ❆ ♠❛tr♦✐❞ M ✐s ❛ ♣❛✐r ( E, I ) ✱ ✇❤❡r❡ E ✐s ❛ s❡t ✭❝❛❧❧❡❞ t❤❡ ❣r♦✉♥❞ s❡t✮ ❛♥❞ I ✐s ❛ ❢❛♠✐❧② ♦❢ s✉❜s❡ts ♦❢ E ✭❝❛❧❧❡❞ t❤❡ ✐♥❞❡♣❡♥❞❡♥t s❡ts✮ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿ ✶ ❊✈❡r② s✉❜s❡t ♦❢ ❛♥ ✐♥❞❡♣❡♥❞❡♥t s❡t ✐s ✐♥❞❡♣❡♥❞❡♥t✱ ✐✳❡✳✱ ❢♦r ❡❛❝❤ A ′ ⊂ A ⊂ E ✱ ✐❢ A ∈ I t❤❡♥ A ′ ∈ I ✭❤❡r❡❞✐t❛r② ♣r♦♣❡rt②✮✳ ✷ ■❢ A ❛♥❞ B ❛r❡ t✇♦ ✐♥❞❡♣❡♥❞❡♥t s❡ts ♦❢ I ❛♥❞ | A | > | B | ✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ❡❧❡♠❡♥t ✐♥ A t❤❛t ✇❤❡♥ ❛❞❞❡❞ t♦ B ❣✐✈❡s ❛ ❧❛r❣❡r ✐♥❞❡♣❡♥❞❡♥t s❡t t❤❛♥ B ✭❡①❝❤❛♥❣❡ ♣r♦♣❡rt②✮✳ ❚❛❦❡ t❤r❡❡ ❧✐♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t ✈❡❝t♦rs E = { v 1 , v 2 , v 3 } ♦❢ s♦♠❡ ✈❡❝t♦r s♣❛❝❡✳ ❚❤❡♥ ✐♥❞❡♣❡♥❞❡♥t s❡ts ♦❢ t❤❡ ♠❛tr♦✐❞ M = ( E, I ) ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ∅ , { v 1 } , { v 2 } , { v 3 } , { v 1 , v 2 } , { v 1 , v 3 } , { v 2 , v 3 } , { v 1 , v 2 , v 3 }

❚❤❡ ❢r❡❡ ♠❛tr♦✐❞ ♦❢ r❛♥❦ n ✐s t❤❡ s✐♠♣❧❡st ♠❛tr♦✐❞ ❡✈❡r✳ ■t ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ❝❧❛ss ♦❢ ❛❧❧ ✐♥❞❡♣❡♥❞❡♥t s❡ts ❣❡♥❡r❛t❡❞ ❜② n ✈❡❝t♦rs✳ Pr♦❜❧❡♠ ❆r❡ t❤❡r❡ ♦t❤❡r s✐♠♣❧❡ ♠❛tr♦✐❞s❄ ■❞❡❛✦ ❖♥❡ ❝❛♥ ❝♦♥s✐❞❡r ♠❛tr♦✐❞s ❛s ▲❘❇✲s t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ♠❛tr♦✐❞s✳

▲❡t M = ( E, I ) ❜❡ ❛ ♠❛tr♦✐❞ ❛♥❞ − → I ❜❡ t❤❡ ❝❧❛ss ♦❢ ❛❧❧ ♦r❞❡r❡❞ s❡ts ♦❢ I ✳ ❉❡✜♥❡ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦✈❡r − → I ❜② ( v 1 , v 2 , . . . , v n )( u 1 , u 2 , . . . , u m ) = ( v 1 , v 2 , . . . , v n , u i 1 , u i 2 , . . . , u i k ) , ✇❤❡r❡ ❡❛❝❤ u i j ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ♣r❡✈✐♦✉s ❡❧❡♠❡♥ts✳ ❚❤✉s✱ t❤❡ ❝❧❛ss ♦❢ ♦r❞❡r❡❞ ✐♥❞❡♣❡♥❞❡♥t s❡ts − → I ❜❡❝♦♠❡s ❛♥ ▲❘❇✳ ❋♦r ❡①❛♠♣❧❡✱ − → I ♦❢ t❤❡ ♠❛tr♦✐❞ ❣❡♥❡r❛t❡❞ ❜② ❧✐♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t ✈❡❝t♦rs v 1 , v 2 , v 3 ✐s ✐s♦♠♦r♣❤✐❝ t♦ F 3 ✳

▲❡t − → I n ❜❡ ❛❧❧ ♦r❞❡r❡❞ ✐♥❞❡♣❡♥❞❡♥t s❡ts ♦❢ t❤❡ ❢r❡❡ ♠❛tr♦✐❞ ♦❢ r❛♥❦ n ✳ − → I n ✐s ✐s♦♠♦r♣❤✐❝ t♦ F n ✳ ❉❡♥♦t❡ t❤❡ ❢r❡❡ ❧❡❢t r❡❣✉❧❛r ❜❛♥❞ ♦❢ ❝♦✉♥t❛❜❧❡ r❛♥❦ ❜② F ✭✐t ❝♦♥t❛✐♥s ❛❧❧ F n ✮✳ ▲❡t ✉s ❣✐✈❡ t❤❡ ❞❡✜♥✐t✐♦♥✿ ✏❛ ✜♥✐t❡ ♠❛tr♦✐❞ M = ( E, I ) ✐s s✐♠♣❧❡ ✐❢ − → I ✐s ❡♠❜❡❞❞❡❞ ✐♥ F ✑✳ ❚❤✉s✱ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ st✉❞② ✜♥✐t❡ s✉❜❜❛♥❞s ♦❢ F ✳

Pr♦♣❡rt✐❡s ♦❢ s✉❜❜❛♥❞s ✐♥ F ❍❛ss❡ ❞✐❛❣r❛♠ ♦❢ t❤❡ ♦r❞❡r ≤ ✐s ❛ tr❡❡ ❢♦r ❛♥② s✉❜❜❛♥❞ S ♦❢ F ✳ ❇❛♥❞s ✇✐t❤ s✉❝❤ ♣r♦♣❡rt② ♦❢ ≤ ❛r❡ ❝❛❧❧❡❞ r✐❣❤t ❤❡r❡❞✐t❛r②✳ ❆r❡ ♦t❤❡r ♣r♦♣❡rt✐❡s❄ ▲❡t α ( x ) ❜❡ t❤❡ ❛♥❝❡st♦r ♦❢ x ∈ S r❡❧❛t✐✈❡ t❤❡ ♦r❞❡r ≤ ✳ ❙✐♥❝❡ t❤❡ ❍❛ss❡ ❞✐❛❣r❛♠ ♦❢ ≤ ✐s ❛ tr❡❡✱ α ( x ) ✐s ✉♥✐q✉❡✳ S c = { s | α ( s ) c = sc, α ( s ) α ( c ) � = sα ( c ) } . ❲❡ s❛② t❤❛t S ❤❛s ❧♦❝❛❧ ❧✐♥❡❛r ♦r❞❡r ✐❢ ❡❛❝❤ s❡t S c ❛❞♠✐ts ❛ ❧✐♥❡❛r ♦r❞❡r ⊏ c s✉❝❤ t❤❛t ✶ ✐❢ S b ⊆ S c ❛♥❞ x ⊏ b y t❤❡♥ x ⊏ c y ❀ ✷ ✈❡r② ❝♦♠♣❧✐❝❛t❡❞ ❝♦♥❞✐t✐♦♥❀ ✸ ❜♦r✐♥❣ ❝♦♥❞✐t✐♦♥✳